71.16.7 problem 7

Internal problem ID [14556]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 5. The Laplace Transform Method. Exercises 5.5, page 273
Problem number : 7
Date solved : Tuesday, January 28, 2025 at 06:43:25 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+a^{2} y&=\delta \left (x -\pi \right ) f \left (x \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 10.466 (sec). Leaf size: 24

dsolve([diff(y(x),x$2)+a^2*y(x)=Dirac(x-Pi)*f(x),y(0) = 0, D(y)(0) = 0],y(x), singsol=all)
 

\[ y = \frac {\operatorname {Heaviside}\left (x -\pi \right ) f \left (\pi \right ) \sin \left (a \left (x -\pi \right )\right )}{a} \]

✓ Solution by Mathematica

Time used: 0.102 (sec). Leaf size: 126

DSolve[{D[y[x],{x,2}]+a^2*y[x]==DiracDelta[x-Pi]*f[x],{y[0]==0,Derivative[1][y][0] ==0}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\sin (a x) \int _1^0\frac {\cos (a \pi ) \delta (\pi -K[2]) f(\pi )}{a}dK[2]+\sin (a x) \int _1^x\frac {\cos (a \pi ) \delta (\pi -K[2]) f(\pi )}{a}dK[2]-\cos (a x) \int _1^0-\frac {\delta (\pi -K[1]) f(\pi ) \sin (a \pi )}{a}dK[1]+\cos (a x) \int _1^x-\frac {\delta (\pi -K[1]) f(\pi ) \sin (a \pi )}{a}dK[1] \]